Mathematics · Intermediate · 40 min · Chapter 6 of 28

Differential calculus

The derivative is the instantaneous rate of change: a limit of slopes. The rules let you compute it without returning to the limit each time.

Read first: Functions, exponentials, and logarithms, Trigonometry

The idea

The average rate of change of f between a and a + h is the slope of the secant: [f(a + h) − f(a)] / h. The derivative is the limit of that quotient as h approaches 0, when the limit exists. Geometrically it is the slope of the tangent. Physically, if f is position, f′ is velocity and f″ is acceleration. The units of a derivative are the units of the output per unit of the input.

You almost never compute that limit by hand after the first week. The power rule says the derivative of xⁿ is n xⁿ⁻¹. The product rule is (fg)′ = f′g + fg′, not f′g′. The quotient rule is the product rule applied to f · g⁻¹. The chain rule handles composition: (g ∘ f)′(x) = g′(f(x)) · f′(x). In words, differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.

A function can change direction only where the derivative is zero or fails to exist. Those are the critical points. The first-derivative test looks at the sign of f′ on either side. The second-derivative test uses concavity: if f′(a) = 0 and f″(a) > 0, the curve is concave up and a is a local minimum. None of this locates a global maximum on an open interval; you still compare values, and on a closed interval you also check the endpoints.

Hold onto this

The derivative f′(a) is the slope of the tangent line at x = a.
Differentiability implies continuity, but not the other way around.
Power, product, quotient, and chain rules reduce derivatives of formulas to algebra.
Critical points, where f′ is zero or undefined, are the candidates for local maxima and minima.

Formulas

Definition of the derivative

f^{'} (a) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}

Instantaneous rate of change at x = a, when this limit exists.

h: A small increment in the input
f′(a): Derivative of f at a

Power rule

\frac{d}{d x} x^{n} = n x^{n - 1}

Bring down the exponent and reduce it by one. It holds for any real n where the expression makes sense.

n: A constant exponent

Product rule

(f g)^{'} = f^{'} g + f g^{'}

Differentiate one factor at a time and add the two results.

f, g: Differentiable functions

Chain rule

\frac{d}{d x} g (f (x)) = g^{'} (f (x)) f^{'} (x)

Differentiate the outer function, then multiply by the derivative of the inner function.

g′(f(x)): Derivative of g, evaluated at f(x)

Exponential and logarithm

\frac{d}{d x} e^{x} = e^{x}, \frac{d}{d x} ln x = \frac{1}{x}

The natural exponential is its own derivative. The logarithm’s derivative is the reciprocal, for x > 0.

eˣ: Exponential base e

Worked examples

1. From the definition, once

Use the definition to differentiate f(x) = x² at a general x.

Difference quotient: [(x + h)² − x²] / h = (x² + 2xh + h² − x²) / h.
Simplify: (2xh + h²) / h = 2x + h, for h ≠ 0.
Limit as h → 0: 2x.

f′(x) = 2x, which matches the power rule.

2. Chain rule with a power

Differentiate y = (3x² − 1)⁴.

Outer function: u⁴, whose derivative is 4u³.
Inner function: u = 3x² − 1, whose derivative is 6x.
Multiply: 4(3x² − 1)³ · 6x = 24x(3x² − 1)³.

dy/dx = 24x (3x² − 1)³.

Try these

Find the critical points of f(x) = x³ − 3x.

Differentiate x eˣ.

Common mistakes

The derivative of a product is not the product of the derivatives.
A corner, such as |x| at 0, can be continuous and still not differentiable.
A critical point is only a candidate. f(x) = x³ has f′(0) = 0 and is neither a max nor a min.